Dynamical symmetry of the Kaluza-Klein monopole∗

The Kepler-type dynamical symmetries of the Kaluza-Klein monopole are reviewed. At the classical level, the conservation of the angular momentum and of a Runge-Lenz vector imply that the trajectories are conic sections. The o(4) algebra allows us to calculate the bound-state spectrum, and the o(3, 1) algebra yields the scattering matrix. The symmetry algebra extends to o(4, 2). 1 SUMMARY OF KALUZA-KLEIN THEORY [1, 2] One of the oldest and most enduring ideas regarding the unification of gravitation and gauge theory is Kaluza’s five dimensional unified theory. Kaluza’s hypothesis was that the world has four spatial dimensions, but one of the dimensions has curled up to form a circle so small as to be unobservable. He showed that ordinary general relativity in five dimensions, assuming such a cylindrical ground state, contained a local U(1) gauge symmetry arising from the isometry of the hidden fifth dimension. The extra components of the metric tensor constitute the gauge fields of this symmetry and could be identified with the electromagnetic vector potential. To be more specific, consider general relativity on a five dimensional space-time with the Einstein-Hilbert action S = − 1 16πGK ∫ dx √ −g5R5 (1.1) where R5 is the five-dimensional curvature scalar of the metric gAB, g5 = det(gAB), andGK is the five-dimensional coupling constant. Our conventions are: upper case Latin letters A,B,C denote five-dimensional indices 0, 1, 2, 3, 5; lower case Greek indices μ, ν . . . run over four dimensions, 0, 1, 2, 3, whereas lower case Latin indices run over four-dimensional spatial values 1, 2, 3, 5. The signature of gAB is (−,+,+,+,+) and the Riemann tensor is RLMN = ∂MΓ K LN − ∂NΓLM + ΓJMΓLN − ΓJNΓLM RLM = RLKM , R5 = R L L, and, except where indicated, h/2π = c = 1. In the absence of other fields the equations of motion are of course RAB = 0. The basic assumption of Kaluza and Klein was that the correct vacuum is the space M×SR, the product of four dimensional Minkowski space with a circle of radius R. The radius of the ∗Talk given at the ’88 Schloss Hofen Meeting on Symmetries in Science III. Gruber B and Iachello F (eds). Plenum : New York. p. 399-417 (1989). †Bolyai Institute, University of Szeged. Present address: Research Institute for Particle and Nuclear Physics, Budapest, Hungary. e-mail: lfeher-at-rmki.kfki.hu ‡Dipartimento di Fisica, Università di Napoli, Italy. Present address: Laboratoire de Mathématiques et de Physique Théorique, Université de Tours, France. e-mail: horvathy-at-lmpt.univ-tours.fr 1 ar X iv :0 90 2. 46 00 v1 [ he pth ] 2 6 Fe b 20 09 circle in the fifth dimension is undetermined by the classical equations of motion, since any circle is flat. If R is sufficiently small then all low-energy experiments will simply average over the fifth dimension. In fact the components of the metric, gAB(x, x5), can be expanded in Fourier series, gAB(x, x) = ∑ n g (n) AB(x ) exp [ inx5 R ] , (1.2) and all modes with n 6= 0 will have energies greater than hc/R. Thus the effective low-energy theory can be deduced by considering the metric gAB to be independent of x5. Under these assumptions the theory is invariant under general coordinate transformations that are independent of x5. In addition to ordinary four dimensional coordinate transformations xμ → xμ(xν), we have a U(1) local gauge transformation x5 → x5 + Λ(xμ), under which gμ5 transforms as a vector gauge field, gμ5(x)→ gμ5(x) + ∂μΛ. (1.3) Therefore the low-energy theory should be a theory of four dimensional gravity plus a U(1) gauge theory, i.e. electromagnetism, with the massless modes of gμν , gμ5 corresponding to the graviton (photon). The low-energy theory is also invariant under scale transformations in the fifth dimension, x → λx, g55 → λg55, gμ5 → λgμ5. (1.4) This global scale invariance is spontaneously broken by the Kaluza-Klein vacuum (since R is fixed) thus giving rise to a Goldstone boson, the dilaton. To exhibit the low-energy theory we write the metric as follows gAB =  gμν +AμAν AμV AνV V  , g5 = detgAB = det(gμν)V = g4V, ds2 = V (dx5 +Aμdx) + gμνdxdx . (1.5) The five-dimensional curvature scalar can be expressed in terms of the four dimensional curvature, R4, the field strength, Fμν = ∂μAν − ∂νAμ, and the scalar field, V , R5 = R4 + 1 4V FμνF μν − 2 √ V √ V . (1.6) Thus the effective low energy theory is described by the four dimensional action S = − 1 16πG ∫ dx √ −g4 V 1/2 ( R4 + 14V FμνF μν ) , (1.7) where we have dropped the terms in R5 involving V , since, when multiplied by √ g5, these yield a total derivative, and G = GK 2πR (1.8) is Newton’s constant, determined by √ hG/ √ 2πc3 ≈ 1.6× 10−33 cm. This theory is recognizable as a variant of the Brans-Dicke theory [3] of gravity, with V 1/2 identified as a Brans–Dicke massless scalar field, coupled to electromagnetism. V indeed sets the local scale of the gravitational coupling. In the vacuum V = 1. Also in the Brans–Dicke theory the coupling of V to matter is somewhat arbitrary, here it is totally fixed by five-dimensional